The Bifurcation Analysis Of Two Types Of Biological Models

Neuron is the component basic unit of the nervous system, the research of individualneurons firing pattern is the basis of the study of the nervous system information transmission.By establishing a mathematical model of the neurons, using the nonlinear dynamics theoryand method for researches, we can more deeply reveals the essence of different firing patternsof neurons activity mechanism, for people to know the function of the nervous system is ofgreat significance.In this paper, using the differential equations qualitative theory and the knowledge ofbifurcation theory, combined with the result of numerical simulation, the models for twoclasses of biological neurons were studied. One is a neuron model of Hodgkin-Huxley type,which describes the changes of muscle cell membrane potential, we analysis the singleparameter bifurcation and two parameters bifurcations of equilibria of this model. For theBogdanov-Takens bifurcation of a equilibrium, we obtain the saddle-node bifurcationcurve, Hopf bifurcation curve and homoclinic bifurcation curve near the BT point. Another isa simplified model of the leech heart interneurons, we analysis the bifurcations of equilibriaand limit cycles with the changing parameters. The period-doubling bifurcations are alsodiscussed.This paper is divided four chapters:The first chapter introduces the meaning and the research status at home and abroad ofthe biological neuron model, as well as the main research work and purpose of this paper.The second chapter describes the basic concepts and research methods.The third chapter discussed the single-parameter bifurcation and two-parameterbifurcation of the Hodgkin-Huxley model which was used to describe the muscle cellmembrane potential changes, and analyse the Bogdanov-Takens branch of the model.The fourth chapter we discussed the leech neuron model, and discussed the dynamicchanges and bifurcations of the model by analyzing the changing of parameters.